Conchoid of Nicomedes

$\begin{figure}\begin{center}\BoxedEPSF{conchoid.epsf scaled 705}\end{center}\end{figure}$

A curve studied by the Greek mathematician Nicomedes in about 200 BC , also called the Cochloid. It is the Locus of points a fixed distance away from a line as measured along a line from the Focus point (MacTutor Archive). Nicomedes recognized the three distinct forms seen in this family. This curve was a favorite with 17th century mathematicians and could be used to solve the problems of Cube Duplication and Angle Trisection.

In Polar Coordinates,

$\begin{displaymath} r=b+a\sec\theta. \end{displaymath}$

(1)

In Cartesian Coordinates,

$\begin{displaymath} (x-a)^2(x^2+y^2)=b^2x^2. \end{displaymath}$

(2)

The conchoid has

as an asymptote and the Area between either branch and the Asymptote is infinite. The Area of the loop is

$\begin{displaymath} A = a\sqrt{b^2-a^2} - 2ab\ln\left({b+\sqrt{b^2-a^2}\over a}\right)+ b^2\cos^{-1}\left({a\over b}\right). \end{displaymath}$

(3)

See also Conchoid

References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 135-139, 1972.

Lee, X. ``Conchoid of Nicomedes.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/ConchoidOfNicomedes_dir/conchoidOfNicomedes.html.

MacTutor History of Mathematics Archive. ``Conchoid.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Conchoid.html.

Pappas, T. ``Conchoid of Nicomedes.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 94-95, 1989.

Yates, R. C. ``Conchoid.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 31-33, 1952.